PPATH - Prime Path

The ministers of the cabinet were quite upset by the message from the Chief of Security stating that they would all have to change the four-digit room numbers on their offices.
— It is a matter of security to change such things every now and then, to keep the enemy in the dark.
— But look, I have chosen my number 1033 for good reasons. I am the Prime minister, you know!
— I know, so therefore your new number 8179 is also a prime. You will just have to paste four new digits over the four old ones on your office door.
— No, it's not that simple. Suppose that I change the first digit to an 8, then the number will read 8033 which is not a prime!
— I see, being the prime minister you cannot stand having a non-prime number on your door even for a few seconds.
— Correct! So I must invent a scheme for going from 1033 to 8179 by a path of prime numbers where only one digit is changed from one prime to the next prime.

Now, the minister of finance, who had been eavesdropping, intervened.
— No unnecessary expenditure, please! I happen to know that the price of a digit is one pound.
— Hmm, in that case I need a computer program to minimize the cost. You don't know some very cheap software gurus, do you?
— In fact, I do. You see, there is this programming contest going on...

Help the prime minister to find the cheapest prime path between any two given four-digit primes! The first digit must be nonzero, of course. Here is a solution in the case above.

The cost of this solution is 6 pounds. Note that the digit 1 which got pasted over in step 2 can not be reused in the last step – a new 1 must be purchased.


One line with a positive number: the number of test cases (at most 100). Then for each test case, one line with two numbers separated by a blank. Both numbers are four-digit primes (without leading zeros).


One line for each case, either with a number stating the minimal cost or containing the word Impossible.


1033 8179
1373 8017
1033 1033


hide comments
rraj001: 2016-05-15 12:21:45

nice problem,AC in one go!!!

dwij28: 2016-04-11 08:26:36

Bfs and Sieve of Eratosthenes.. :) Time limit and test cases are very lenient so it passes even if algorithm for checking single digit difference takes 0(n^2), where n is the no. of 4 digit primes..

karthik1997: 2016-03-23 14:33:41

Simple Sieve and a BFS . Ac in one go . Learnt the importance of implementation rather than the algo :P

hardik agrawal: 2016-02-27 13:36:20

nice problem :D

dhumketu: 2016-01-27 15:51:46

Literally Nothing is IMPOSSIBLE!! :D Didn't even consider a possibility of impossible.

Abhishek Naik: 2016-01-24 14:29:24

One important thing that I noticed - The next number in the sequence can be smaller than the previous one. For e.g., my accepted program gives the following output for the 1st test case (printed in the reverse order due to backtracking):
This is different than the sequence that is given in the example above.

Deepak : 2016-01-19 14:41:34

nice one..AC in one go.

anandrohit: 2016-01-15 12:09:25

Importance of < & <=
bheja fry

Last edit: 2016-01-15 12:09:46
lakshay_v06: 2016-01-14 15:41:50

Gr8 ques! Basic graph theory + sieve ! Done in 0.00! :D

abhinav_rai: 2015-12-17 23:00:29

Cool Problem. Made me remember bfs and heights along with sieve and how to check for single digit change. Indeed great problem!!

Added by:overwise
Time limit:2s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ERL JS-RHINO NODEJS PERL6 VB.NET
Resource:ACM ICPC NWERC 2006